Had it not been for Dellin Betances giving up a home run to the Red Sox’s Mookie Betts in the seventh inning Wednesday night, Alex Rodriguez’s sixth-inning blast to left would have carried the Yankees to a wild-card clinching victory at Yankee Stadium.

Instead, Betts’ homer tied the score and four runs in the 11th lifted the Red Sox to a 9-5 victory.

Nevertheless, Rodriguez’s first homer since Sept. 15 was an indication the designated hitter could be emerging from the 9-for-45 (.200) slump he lugged into the game.

“I thought he swung the bat extremely well,’’ manager Joe Girardi said of Rodriguez, who singled in the third and walked in the fifth and eighth. It was the first time since Aug. 6 he reached base more than twice in a game. “[Tuesday] was a long emotional day for a lot of guys. We all went to Yogi’s funeral. I don’t know if that had anything to do with it, but he looked great today.’’

Rodriguez’s 33rd homer, on Wednesday, produced his 1,065th RBI as a Yankee and tied him with Jorge Posada for 11th place on the all-time franchise list.

Without Mark Teixeira, the Yankees are susceptible to left-handed pitching and need right-handed bats such as Rodriguez and switch-hitters Carlos Beltran and Chase Headley to produce.

“There is some concern of fatigue, of course,’’ Girardi said of the 40-year-old Rodriguez, who played in his 147th game (136th start) of the season. “But we need him now and that’s the bottom line. If we can clinch our [wild-card] spot, I can give him a few days off.

That’s the hope that we can deal with some of our regulars but we haven’t been able to do it. I gave him a day off Sunday hoping that would help. We are trying to wrap this up, so I can take care some of these guys.’’

Stephen Drew was examined in Pittsburgh by a concussion specialist.

Stephen DrewRay Stubblebine

“That’s because that is the [doctor] he saw before,’’ Girardi said of the infielder, who suffered a concussion in 2013. “I am assuming he will be back [Thursday].’’

There is a possibility Drew won’t be available for postseason play.

There is little doubt pinch-running specialist Rico Noel will be on the Yankees’ roster for the expected AL wild-card game Tuesday. If the Yankees advance to the ALDS, they will have to decide if the luxury is worth a roster spot.

“I think you definitely have to consider it,’’ Girardi said of carrying Noel, who wasn’t much at the plate in the minor leagues and has not played a lot of outfield since being recalled from Triple-A Scranton/Wilkes-Barre. “I haven’t talked about it because it’s premature.’’

Nevertheless, with Rodriguez, Brian McCann and Beltran in the lineup the need for Noel, who is 5-for-5 in stolen bases after a steal of second in the seventh inning, could be a game-changer.

Girardi is trying to decide if he needs a third lefty reliever behind Andrew Miller and Justin Wilson during the postseason.

“You are trying to figure out who is the best guy to use in that situation,’’ he said. “[James] Pazos has the velocity, [Chris] Capuano has the assortment and [Chasen] Shreve has the splitter. We are trying to see if we can get somebody who can get left-handers out.’’

Shreve didn’t help his cause by giving up a two-run homer to Betts in the 11th inning.